Integrand size = 13, antiderivative size = 115 \[ \int \frac {x^{2/3}}{(a+b x)^2} \, dx=-\frac {x^{2/3}}{b (a+b x)}-\frac {2 \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} \sqrt [3]{a} b^{5/3}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{\sqrt [3]{a} b^{5/3}}+\frac {\log (a+b x)}{3 \sqrt [3]{a} b^{5/3}} \]
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Time = 0.03 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {43, 58, 631, 210, 31} \[ \int \frac {x^{2/3}}{(a+b x)^2} \, dx=-\frac {2 \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} \sqrt [3]{a} b^{5/3}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{\sqrt [3]{a} b^{5/3}}+\frac {\log (a+b x)}{3 \sqrt [3]{a} b^{5/3}}-\frac {x^{2/3}}{b (a+b x)} \]
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Rule 31
Rule 43
Rule 58
Rule 210
Rule 631
Rubi steps \begin{align*} \text {integral}& = -\frac {x^{2/3}}{b (a+b x)}+\frac {2 \int \frac {1}{\sqrt [3]{x} (a+b x)} \, dx}{3 b} \\ & = -\frac {x^{2/3}}{b (a+b x)}+\frac {\log (a+b x)}{3 \sqrt [3]{a} b^{5/3}}+\frac {\text {Subst}\left (\int \frac {1}{\frac {a^{2/3}}{b^{2/3}}-\frac {\sqrt [3]{a} x}{\sqrt [3]{b}}+x^2} \, dx,x,\sqrt [3]{x}\right )}{b^2}-\frac {\text {Subst}\left (\int \frac {1}{\frac {\sqrt [3]{a}}{\sqrt [3]{b}}+x} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{a} b^{5/3}} \\ & = -\frac {x^{2/3}}{b (a+b x)}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{\sqrt [3]{a} b^{5/3}}+\frac {\log (a+b x)}{3 \sqrt [3]{a} b^{5/3}}+\frac {2 \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt [3]{a}}\right )}{\sqrt [3]{a} b^{5/3}} \\ & = -\frac {x^{2/3}}{b (a+b x)}-\frac {2 \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt {3} \sqrt [3]{a} b^{5/3}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{\sqrt [3]{a} b^{5/3}}+\frac {\log (a+b x)}{3 \sqrt [3]{a} b^{5/3}} \\ \end{align*}
Time = 0.15 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.16 \[ \int \frac {x^{2/3}}{(a+b x)^2} \, dx=\frac {-\frac {3 b^{2/3} x^{2/3}}{a+b x}-\frac {2 \sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt [3]{a}}-\frac {2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{\sqrt [3]{a}}+\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sqrt [3]{x}+b^{2/3} x^{2/3}\right )}{\sqrt [3]{a}}}{3 b^{5/3}} \]
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Time = 0.27 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.03
method | result | size |
derivativedivides | \(-\frac {x^{\frac {2}{3}}}{b \left (b x +a \right )}+\frac {-\frac {2 \ln \left (x^{\frac {1}{3}}+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (x^{\frac {2}{3}}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x^{\frac {1}{3}}+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x^{\frac {1}{3}}}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}}{b}\) | \(118\) |
default | \(-\frac {x^{\frac {2}{3}}}{b \left (b x +a \right )}+\frac {-\frac {2 \ln \left (x^{\frac {1}{3}}+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (x^{\frac {2}{3}}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x^{\frac {1}{3}}+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x^{\frac {1}{3}}}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}}{b}\) | \(118\) |
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Leaf count of result is larger than twice the leaf count of optimal. 175 vs. \(2 (84) = 168\).
Time = 0.24 (sec) , antiderivative size = 394, normalized size of antiderivative = 3.43 \[ \int \frac {x^{2/3}}{(a+b x)^2} \, dx=\left [-\frac {3 \, a b^{2} x^{\frac {2}{3}} - 3 \, \sqrt {\frac {1}{3}} {\left (a b^{2} x + a^{2} b\right )} \sqrt {\frac {\left (-a b^{2}\right )^{\frac {1}{3}}}{a}} \log \left (\frac {2 \, b^{2} x - a b + 3 \, \sqrt {\frac {1}{3}} {\left (a b x^{\frac {1}{3}} + \left (-a b^{2}\right )^{\frac {1}{3}} a + 2 \, \left (-a b^{2}\right )^{\frac {2}{3}} x^{\frac {2}{3}}\right )} \sqrt {\frac {\left (-a b^{2}\right )^{\frac {1}{3}}}{a}} - 3 \, \left (-a b^{2}\right )^{\frac {2}{3}} x^{\frac {1}{3}}}{b x + a}\right ) - \left (-a b^{2}\right )^{\frac {2}{3}} {\left (b x + a\right )} \log \left (b^{2} x^{\frac {2}{3}} + \left (-a b^{2}\right )^{\frac {1}{3}} b x^{\frac {1}{3}} + \left (-a b^{2}\right )^{\frac {2}{3}}\right ) + 2 \, \left (-a b^{2}\right )^{\frac {2}{3}} {\left (b x + a\right )} \log \left (b x^{\frac {1}{3}} - \left (-a b^{2}\right )^{\frac {1}{3}}\right )}{3 \, {\left (a b^{4} x + a^{2} b^{3}\right )}}, -\frac {3 \, a b^{2} x^{\frac {2}{3}} - 6 \, \sqrt {\frac {1}{3}} {\left (a b^{2} x + a^{2} b\right )} \sqrt {-\frac {\left (-a b^{2}\right )^{\frac {1}{3}}}{a}} \arctan \left (\frac {\sqrt {\frac {1}{3}} {\left (2 \, b x^{\frac {1}{3}} + \left (-a b^{2}\right )^{\frac {1}{3}}\right )} \sqrt {-\frac {\left (-a b^{2}\right )^{\frac {1}{3}}}{a}}}{b}\right ) - \left (-a b^{2}\right )^{\frac {2}{3}} {\left (b x + a\right )} \log \left (b^{2} x^{\frac {2}{3}} + \left (-a b^{2}\right )^{\frac {1}{3}} b x^{\frac {1}{3}} + \left (-a b^{2}\right )^{\frac {2}{3}}\right ) + 2 \, \left (-a b^{2}\right )^{\frac {2}{3}} {\left (b x + a\right )} \log \left (b x^{\frac {1}{3}} - \left (-a b^{2}\right )^{\frac {1}{3}}\right )}{3 \, {\left (a b^{4} x + a^{2} b^{3}\right )}}\right ] \]
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Leaf count of result is larger than twice the leaf count of optimal. 527 vs. \(2 (107) = 214\).
Time = 135.94 (sec) , antiderivative size = 527, normalized size of antiderivative = 4.58 \[ \int \frac {x^{2/3}}{(a+b x)^2} \, dx=\begin {cases} \frac {\tilde {\infty }}{\sqrt [3]{x}} & \text {for}\: a = 0 \wedge b = 0 \\\frac {3 x^{\frac {5}{3}}}{5 a^{2}} & \text {for}\: b = 0 \\- \frac {3}{b^{2} \sqrt [3]{x}} & \text {for}\: a = 0 \\\frac {2 a \log {\left (\sqrt [3]{x} - \sqrt [3]{- \frac {a}{b}} \right )}}{3 a b^{2} \sqrt [3]{- \frac {a}{b}} + 3 b^{3} x \sqrt [3]{- \frac {a}{b}}} - \frac {a \log {\left (4 x^{\frac {2}{3}} + 4 \sqrt [3]{x} \sqrt [3]{- \frac {a}{b}} + 4 \left (- \frac {a}{b}\right )^{\frac {2}{3}} \right )}}{3 a b^{2} \sqrt [3]{- \frac {a}{b}} + 3 b^{3} x \sqrt [3]{- \frac {a}{b}}} + \frac {2 \sqrt {3} a \operatorname {atan}{\left (\frac {2 \sqrt {3} \sqrt [3]{x}}{3 \sqrt [3]{- \frac {a}{b}}} + \frac {\sqrt {3}}{3} \right )}}{3 a b^{2} \sqrt [3]{- \frac {a}{b}} + 3 b^{3} x \sqrt [3]{- \frac {a}{b}}} + \frac {2 a \log {\left (2 \right )}}{3 a b^{2} \sqrt [3]{- \frac {a}{b}} + 3 b^{3} x \sqrt [3]{- \frac {a}{b}}} - \frac {3 b x^{\frac {2}{3}} \sqrt [3]{- \frac {a}{b}}}{3 a b^{2} \sqrt [3]{- \frac {a}{b}} + 3 b^{3} x \sqrt [3]{- \frac {a}{b}}} + \frac {2 b x \log {\left (\sqrt [3]{x} - \sqrt [3]{- \frac {a}{b}} \right )}}{3 a b^{2} \sqrt [3]{- \frac {a}{b}} + 3 b^{3} x \sqrt [3]{- \frac {a}{b}}} - \frac {b x \log {\left (4 x^{\frac {2}{3}} + 4 \sqrt [3]{x} \sqrt [3]{- \frac {a}{b}} + 4 \left (- \frac {a}{b}\right )^{\frac {2}{3}} \right )}}{3 a b^{2} \sqrt [3]{- \frac {a}{b}} + 3 b^{3} x \sqrt [3]{- \frac {a}{b}}} + \frac {2 \sqrt {3} b x \operatorname {atan}{\left (\frac {2 \sqrt {3} \sqrt [3]{x}}{3 \sqrt [3]{- \frac {a}{b}}} + \frac {\sqrt {3}}{3} \right )}}{3 a b^{2} \sqrt [3]{- \frac {a}{b}} + 3 b^{3} x \sqrt [3]{- \frac {a}{b}}} + \frac {2 b x \log {\left (2 \right )}}{3 a b^{2} \sqrt [3]{- \frac {a}{b}} + 3 b^{3} x \sqrt [3]{- \frac {a}{b}}} & \text {otherwise} \end {cases} \]
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Time = 0.30 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.04 \[ \int \frac {x^{2/3}}{(a+b x)^2} \, dx=-\frac {x^{\frac {2}{3}}}{b^{2} x + a b} + \frac {2 \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, x^{\frac {1}{3}} - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3 \, b^{2} \left (\frac {a}{b}\right )^{\frac {1}{3}}} + \frac {\log \left (x^{\frac {2}{3}} - x^{\frac {1}{3}} \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{3 \, b^{2} \left (\frac {a}{b}\right )^{\frac {1}{3}}} - \frac {2 \, \log \left (x^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \, b^{2} \left (\frac {a}{b}\right )^{\frac {1}{3}}} \]
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Time = 0.30 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.18 \[ \int \frac {x^{2/3}}{(a+b x)^2} \, dx=-\frac {2 \, \left (-\frac {a}{b}\right )^{\frac {2}{3}} \log \left ({\left | x^{\frac {1}{3}} - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{3 \, a b} - \frac {x^{\frac {2}{3}}}{{\left (b x + a\right )} b} - \frac {2 \, \sqrt {3} \left (-a b^{2}\right )^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, x^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3 \, a b^{3}} + \frac {\left (-a b^{2}\right )^{\frac {2}{3}} \log \left (x^{\frac {2}{3}} + x^{\frac {1}{3}} \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{3 \, a b^{3}} \]
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Time = 0.25 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.23 \[ \int \frac {x^{2/3}}{(a+b x)^2} \, dx=\frac {2\,\ln \left (\frac {4\,x^{1/3}}{b}-\frac {4\,{\left (-a\right )}^{1/3}}{b^{4/3}}\right )}{3\,{\left (-a\right )}^{1/3}\,b^{5/3}}-\frac {x^{2/3}}{b\,\left (a+b\,x\right )}+\frac {\ln \left (\frac {4\,x^{1/3}}{b}-\frac {{\left (-a\right )}^{1/3}\,{\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}^2}{b^{4/3}}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{3\,{\left (-a\right )}^{1/3}\,b^{5/3}}-\frac {\ln \left (\frac {4\,x^{1/3}}{b}-\frac {{\left (-a\right )}^{1/3}\,{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^2}{b^{4/3}}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{3\,{\left (-a\right )}^{1/3}\,b^{5/3}} \]
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